Question

given: ( mangle aeb = 45^circ )
( angle aec ) is a right angle.
prove: ( overrightarrow{eb} ) bisects ( angle aec ).
diagram: point ( e ) with ( angle aec ) as right angle, ( a ) left of ( e ), ( c ) above ( e ), ( b ) on a ray from ( e ) to ( b )
proof:
we are given that ( mangle aeb = 45^circ ) and ( angle aec ) is a right angle. the measure of ( angle aec ) is ( 90^circ ) by the definition of a right angle. applying the dropdown options: segment addition property, angle addition postulate, reflexive property, transitive property gives ( mangle aeb )... (remaining text as shown)

Explanation:

Step1: Recall Angle Addition Postulate

The angle addition postulate states that if a point \( B \) lies in the interior of \( \angle AEC \), then \( m\angle AEB + m\angle BEC = m\angle AEC \).

Step2: Apply Given Information

We know \( m\angle AEB = 45^\circ \) and \( m\angle AEC = 90^\circ \) (since \( \angle AEC \) is a right angle). Using the angle addition postulate: \( 45^\circ + m\angle BEC = 90^\circ \).

Step3: Solve for \( m\angle BEC \)

Subtract \( 45^\circ \) from both sides: \( m\angle BEC = 90^\circ - 45^\circ = 45^\circ \).

Step4: Determine Angle Bisector

Since \( m\angle AEB = m\angle BEC = 45^\circ \), by the definition of an angle bisector (a ray that divides an angle into two congruent angles), \( \overrightarrow{EB} \) bisects \( \angle AEC \). The key postulate used in the proof (to relate the angles \( \angle AEB \), \( \angle BEC \), and \( \angle AEC \)) is the angle addition postulate.

Answer:

angle addition postulate